Macromolecules 1995,28, 197-210
197
Swelling and Elastic Properties of Polyelectrolyte Gels R. Skouri, F. Schosseler, J. P. Munch, and S. J. Candau" Laboratoire &Ultrasons et de Dynamique des Fluides Complexes, URA 851, Universitk Louis Pasteur, 4 rue Blaise Pascal, 67070 Strasbourg Cedex, France Received July 22, 1994; Revised Manuscript Received September 26, 1994@
ABSTRACT: Ionized poly(acry1ic acid)gels were studied both at concentrationsclose to the concentration of preparation and at swelling equilibrium. In the first experimental condition, the introduction of electrostatic interactions decreases the shear modulus. The addition of salt screens these interactions and allows one to recover the shear modulus of unneutralized gels. The correlation of these effects with light scattering results suggests that they are related to a change of the gel microstructure with electrostatic interactions. The swelling equilibrium of these gels if found to scale like the ratio of the ionization degree to the Debye-Huckel screening parameter with an exponent 6 6 . The shear modulus at swelling equilibrium is given by the simple affine deformation law for not too high swelling degrees (> a$. In this limit, the properties of the gel depend on the screening range. (i)Weak Screening Regime (KR,< 1). This regime is characterized by a uniform distribution of ions
(13)
Barrat et al. have also studied the collective diffusion in the gel in the long-wavelength limit by using a twofluid model. The collective diffusion De at the swelling equilibrium is given by9 P3,2
-1
2 -115
De = IlO
(14)
where 70is the viscosity of the solvent. Combining eqs 12-14 leads to (15)
(ii) Strong Screening Regime (KR. > 1). In this regime, both the monomers and the counterions are not uniformly distributed throughout the gel. One can consider the chains as surrounded by a sheath of
Macromolecules, Vol. 28, No. 1, 1995
Elastic Properties of Polyelectrolyte Gels 201
countercharges of thickness K - ~ (Figure 4b). In that case, the contribution to the swelling pressure arising from the fluctuations in the monomer density can become nonnegligible as compared to that due to the Donnan equilibrium. In the limit of high ionic strength, the properties of the gel are similar t o those of a usual Flory gel with an electrostatic excluded volume24
12000 3 0
f=OC=IIIM 1=035 C = I M
9000 -
I
h
id
%
c
6000
3.
*
c 1)
3000
I
For such a gel the equilibrium swelling degree is given by the C* (17)
Figure 6. Variation with cross-linking degree of the shear modulus for two gels with different ionization degrees and close to their concentration in the reaction bath. 6ooo
-2
B
It can be noted that relation 18 coincides with relation 12 even though they describe different physical situations represented in Figure 4a and Figure 4b, respectively. It could be argued that the situation encountered in the case of high ionic strength (Figure 4b) mimics on a more microscopic scale the Donnan equilibrium that plays the major role in the weak screening limit. Such an analogy was proposed earlier29and might explain the identity of eqs 12 and 18. Moreover, eq 10 can be recovered easily by setting = 0 into the expression for K in eq 18. This suggests that the latter equation could also be a good approximation to describe intermediate salt concentration conditions. Several remarks can be made with respect to the above statements. (i) For weakly charged gels, one cannot neglect the usual virial contribution to the osmotic pressure so that the effective excluded volume becomes for a 0 solvent v,ff = v,
+ 2vz, + 3w4
1 0 l!t
j-44000$
1
2000
1 1
6 Q C
-
‘0.0
.
eo02
,
’
L?o
0 001
rc
0.2
’
0:4
’
+
+
O
S
0.6
0.8
I ]
1
1.0
a
Figure 6. Influence of neutralization on the shear modulus for two series of gels with different cross-linking degrees ionized after the gelation (C = 1M,C, = 0 M).
The above relationship is valid for ionic strengths small enough that the dominant contribution to the osmotic pressure results from the electrostatic excluded volume. Within this approximation, the elastic moduli and the cooperative diffusion coefficient are given by
(19)
where v is the excluded volume of the neutral polymer, w is the third virial coefficient, and z, is the reduced temperature with respect t o the 0 temperature. (ii) In the presence of a large excess of salt, typically when K - ~ < Zg, the Debye-Hiickel theory is no longer valid and one should recover the behavior of a neutral polymer. As most of the polyelectrolytes have a hydrophobic backbone, one should observe a @ solvent behavior. (iii)Although eq 17 is the same as for neutral gels, it must be kept in mind that as long as we are not in the preceding limit of a very large excess of salt, the excluded volume parameter v e depends on the polymer concentration through eq 16. Therefore, the concentration dependences of the thermodynamic parameters are different for neutral and charged gels. Expressions for the elastic moduli can be derived from the osmotic pressure. If we assume that the Donnan equilibrium is the dominant mechanism, nosis given by9
which, in the limit q5s >> a#,can be approximated by
(23)
which are equivalent to (13) and (15) in the limit
>>
a$. Experimental Results A. Gels in the Reaction Bath. We consider here both nonneutralized samples at the concentration of preparation and samples at a slightly lower concentration prepared by means of the procedure described in the Experimental Section in order to neutralize the gels or to incorporate some salt. Figure 5 shows the variation of the shear modulus with re for quasi-neutral gels (a = 7.5 x and gels with an ionization degree a = 0.35. The shear modulus of quasi-neutral gels is found to be significantly larger than that of the charged ones, except at high cross-link density where there is only a slight difference that can be accounted for simply by the small concentration difference. The effect of the ionization degree on the shear modulus is illustrated in Figure 6 relative to two series of gels with different cross-link densities. Obviously, this effect has to deal with electrostatic interactions since the screening of these interactions by addition of salt produces an increase of the shear modulus that tends to recover the value of the quasi-neutral gels
Macromolecules, Vol. 28, No. 1, 1995
202 Skouri et al. 4000 _--__--
3000
r
9
h
. . -
# 5
-- I
- -a= 0 35
1
Ca=0.171 M
t
1
400
.
2000 L' Y
ua
1000 0
200
(18
-~
00
02
04
06
OR
CE (M)
Figure 7. Influence of salt addition on the shear modulus for two senes of gels wth different neutralization degrees. Gels are first neutralized and then swollen in salted solutions (final concentration C = 0.916 M). Cross-linkingratio is constant.
rc %
Figure 9. Swelling equilibrium ratio versus cross-linking ratio. Gels are ionized after the reaction (Table 1).
rL= 0 02
4000
-_~
___~
-
h
m
4
..
P
I
2000
i
a= 0.35 C5=0.171 M
I
l
O
Figure 8. Effect of the ionization degree on the frozen (closed symbols) and fluctuating (open symbols) contributions t o the scattering intensity. Squares: gels neutralized prior t o the gelation (C = 1.11M, P, = 0.01). Circles: gels neutralized after the gelation (C = 1 M, rc = 0.02). Ionization degrees are
calculated without taking into account Manning's condensation at high neutralization degree (see text).
(Figure 7). Results similar to those of Figure 6 have been reported by Ilmain et al. for both gels prepared from ionized monomers and gels subsequently ionized after preparation, thus demonstrating that this effect is not simply due to a change of network topology with the ionization degree of the m o n ~ m e r s . ~ ~ ~ ~ ~ In Figure 8 are plotted the frozen component, I C ,and the fluctuating contribution, IF,for two gels with different rc as a function of the neutralization degree f. One of the gels (rc= 0.01) was synthesized with preneutralized monomers, while the second one (rc = 0.02) was neutralized after the gelation. In both cases the frozenin contribution is reduced by an increase of the ionization degree. It is not possible from these results to draw a conclusion on the relative effect of the neutralization prior to or after the gelation because polymer concentration and cross-link density are different. A recent study by small-angle neutron scattering (SANS) suggests that the effect of ionization degree on the frozen-in fluctuations is enhanced if the neutralization is performed prior to the gelation.31 Let us mention also that IC is enhanced by an increase of rc.31 In the case of small cross-linking degree, the frozen contribution is found to become smaller than the fluctuating component for large enough ionization degrees. B. Gels at Swelling Equilibrium. Figure 9 shows the effect of the cross-link density on the equilibrium swelling degree Qe of gels prepared a t a polymer concentration C = 1.11M, subsequently neutralized to reach an ionization degree a = 0.35. Salt concentration is kept equal to C, = 0.171 M (1g/L). In Figure 10 is reported the variation of the shear modulus pe of these gels a t the swelling equilibrium as a function of rc.
i
0
1
-
2
1 I
_ . .
3
rc %
4
5
6
Figure 10. Shear modulus at swelling equilibrium versus
cross-linkingratio for the same samples as in Figure 9.
Table 1. Equilibrium Properties of Gels Neutralized after Synthesis (f= 0.35, C, = 0.171 M):Effect of Cross-Link Density 0.5 1 2 3 4
5
472 302 216 148 110 95
240 600 2120 2780 2900 3250
1.0 1.41 3.5 4.1 5.8 10
0.45 0.53 0.59 0.67
19.9 18.6 17.5 16 14.7 14
Qualitatively, one observes the same trends as in the neutral gels with respect to the effect of rc on &e and The results of the experiments performed on gels with different ionization degrees and salt contents are given in Tables 2-4. The general behavior is that previously r e p ~ r t e d . l ~In J ~particular, ,~~ the equilibrium swelling degree increases upon increasing a or decreasing the salt content. This behavior will be discussed in the next section. Turning now to the dynamic properties, we have first proceeded t o a comparison between the kinetics of swelling and dynamic light scattering experiments. One of the problems encountered concerns the analysis of the dynamic light scattering experiments in gels. Two schemes have been considered up to now. The first one assumes that the frozen-in fluctuations arise from the presence of static inhomogeneities whereas the dynamic field associated with long-wavelength dynamic density fluctuations is of same nature as in polymer solution^.^^-^^ In that case, the intensity autocorrelation function can be treated in terms of a heterodyne mixing between these two components. For neutral gels, this leads t o values of De close to those obtained in semidilute solutions.27 Also a good agreement was found between the results of dynamical light scattering obtained using this analysis and those of kinetics of swelling.20
Elastic Properties of Polyelectrolyte Gels 203
Macromolecules, Vol. 28, No.1, 1995 Table 2. Equilibrium Properties of Gels Neutralized after Synthesis (ro= 0.02): Effect of Ionization and Salt Content 0.1
0.2
0.35
360 170 110 70 46 30 22 10 450 210 150 106 70 50 38 16 9 980 480 350 240 150 108 75 40 13
0 0.2 0.5 1 2.5 5 10 25 0 0.2 0.5 1 2.5 5 10 25 50 0 0.2 0.5 1 2.5 5 10 25 50
3600 2800 2900 3300 3800 4400 4900 6900 3900 2700 2600 2800 3200 3600 4200 5500 6900 4100 2800 2500 2300 2300 2500 3000 4200 6400
0.45 0.45 0.45 0.47 0.5 0.48 0.5
4.6 2.2 1.5 1 0.8 0.4 0.33 12 6.9 5.4 3.6 2.2 1.3 0.8 0.4
0.45 0.45 0.45 0.45 0.46 0.46 0.45 0.46
16 11.2 8 4.6 3.4 3 2.1 1.1
0.43 0.44 0.45 0.47 0.45 0.48 0.5 0.53
64.4 33.6 24.1 17.9 12.3 9.1 6.7 4.3 50.9 28.9 21.8 16.9 11.8 8.8 6.6 4.2 3 56.8 31.5 23.6 17.9 12.2 9.1 6.6 4.2 3
Table 3. Swelling Equilibrium Degree of Gels Neutralized after Synthesis (rc= 0.01): Effect of Ionization and Salt Concentration
c,
f = 0.1
f = 0.2
f = 0.3
a
i 0.8 0.6
0.4
I
I
&
-1
n
" 0
20
60
40
Ii
b
800
L 0 ' " " 0
20
40
60
*i
Figure 11. (a) Variation of the amplitude of the intensity autocorrelation function as a function of the scattering intensity when scanning different positions in a gel with C = 1.11 M,f = 0, and r, = 0.02. (b)Variation of the decay time of the intensity autocorrelation function as a function of the scattering intensity (same conditions as in (a)).
f = 0.4
We measured the time-averaged intensity correlation function gT(2)(2) over a set of 997 scattering volume 3970 107 2680 101 1790 102 0 1205 118 positions with an aperture of the detector such that /3 51.9 50.3 1890 50.6 1250 850 54.3 0.1 572 = 0.96 for a gel with C = 1.11M, a = 7.5 x and 1067 37.8 840 38.2 570 38.4 41 0.2 390 rc = 0.02. When scanning over various positions i, the 875 27.4 560 26.7 26.8 380 28.1 0.5 260 normalized functions gd2)(2)can be approximated by 1 625 20.3 377 19.8 19.9 20.6 260 175 1 13.5 13.3 410 13.7 255 173 102 13.5 2.5 6j exp(-thi), where 6i decreases upon increasing the 260 9.7 166 9.6 9.5 9.7 100 5 66 intensity Ij scattered by the corresponding scattering 6.9 6.8 150 90 70 6.9 60 7 10 volume. Figure 11shows the variations of 6i and t i as 90 4.5 70 4.5 55 4.5 4.5 25 37 a function of Ii. It can be seen that zi varies between a 3.2 3.2 66 3.2 53 43 25 3.2 50 lower value 2, for the lowest values of Ii to a higher value of 22, as Ii becomes large. These two limits correspond Table 4. Swelling Equilibrium Degree of Gels Prepared from Neutralized Monomers (rc= 0.005): Effect of to dark and bright speckles, respectively. The same Ionization and Salt Content behavior is observed for all scattering angles investica f = 0.2 f = 0.3 f = 0.4 f = 0.5 gated (20" 5 6 I135"). (EL)&e K - ~ ( A ) Qe K - ~ ( A ) &e K - ~ ( A ) Qe K - ~ ( A ) A critical test to discriminate between the heterodyne mixing and the nonergodic hypothesis is the angular 110 4407 106 115 3760 131 3050 0 2640 2233 52.7 53 54.1 1790 56.6 1464 0.1 1200 dependence of the contributions IFand Ic.35 In our 27.8 1090 27.9 28.3 830 0.5 460 28.2 695 experiments these two contributions are independent 580 19.9 20 20.1 470 20.5 360 1 280 of scattering angle within experimental accuracy, which 9.7 9.7 270 9.7 204 5 120 9.8 170 favors the hypothesis of heterodyne mixing of the 10 75 7 108 7 150 7 194 7 fluctuating component by the static intensity scattered 50 26 3.2 60 3.2 96 3.2 170 3.3 100 4.7 2.1 15 2.2 30 2.3 64 2.3 from small regions ( ~ the ~ agreement is quite satisfactory. Thus kinetics of fact, it seems that this low value could reflect as well swelling experiments are well suited to measure the an averaging of fast cooperative fluctuations in polymer diffusion constant at swelling equilibrium, i.e., when concentration and of very slow motion of inhomogenelight scattering requires tedious matching of gel sizes. Furthermore, the experiments of kinetics of swelling ities. However, the model is not yet suited t o account provide an estimate of the ratio pJM,, (cf. Experimental for that effect.
(a) &e
K-~(A)
Qe
K-~(A)
Qe
K-~(A)
&e
K-'(A)
+
~~~
Macromolecules, Vol. 28, No. 1, 1995
204 Skouri et al. I 2 io5
e
a..03S
h
i
nv
i=o4 varying C,
’
f=O 8, varymg Cs b
cs=o 1 7 1
80106
v
r
Yarylng f, c,= 0
L
40104[
A
0010Oo
I
r,
c
i
3
-
i
4
5
6
1
rc %
f/K
(A)
lo
Figure 12. Influence of cross-linkingdegree on the cooperative diffusion coefficient measured by the kinetics of swelling (data from Table 1)
Figure 13. Variation of the shear modulus with the parameter d ~ Gels . ionized after the reaction. Squares: gels containing added salt (C = 0.916 M). Diamonds: gels without added salt (C = 1 M). Cross-linking ratio is constant: re =
Section). Measured values of De and p$M,, are reported in Tables 1, and 2. Qualitatively, the effect of the cross-linking degree on De is the same as for neutral gels; namely, De increases with rc (Figure 12). Also, the behavior upon a variation of ionization degree andor salt content is that already reported for gels in the reaction bath, i.e., an increase of De upon increasing a andor decreasing Cs.12,30
0.01.
Discussion A. Gels in the Reaction Bath. According to eq 5 , for such gels one would expect the shear modulus to be proportional to the cross-link density. Therefore, the shear modulus at a given polymer concentration should be proportional to rc if the cross-linkers were randomly distributed throughout the sample. This is not what is observed in Figure 5 for quasi-neutral gels. In fact, this is not surprising since it is well known for neutral gels that the occurrence of topological defects like trapped entanglements and free dangling chains can modify the behavior of the shear modulus. Moreover, it has been shown that the gels prepared by radical copolymerization exhibit submicroscopic inhomogeneities whose effect on the thermodynamic properties of gels is not totally elucidated. More surprising is the decrease of the shear mbdulus upon increasing the ionization degree (Figure 6) and the subsequent increase of p as electrostatic interactions are progressively screened by addition of salt (Figure 7). Since the cross-linking degree remains constant in these experiments, the explanation for these behaviors is t o be found in more subtle effects. A possible explanation is provided by the model of Panyukov and Obukhov et al., which states that Ro must be taken as the end-to-end distance the strand would have in a solution at the same concentration.lOJ1 Thus, Ro2 should increase with a because of the swelling of the chains by electrostatic interactions. The mean square end-to-end distance of a chain swollen by the electrostatic excluded volume ue in a semidilute solution isz6
Assuming that the average position of the cross-links is not affected by a variation of a so that R2 = Nu2 remains unchanged, one finds using eqZ5
Figure 13 shows the variation of p as a function of 5 0.4 one can assume f = a. On the other hand, for the data obtained on the sample with f = 0.8, a / would ~ be smaller than f l due ~ to Manning condensation. Indeed inspection of the data in Figure 6 shows that increasing neutralization has little effect on the shear modulus once f L 0.4. A similar effect is observed for the influence of neutralization degree on the two components of the scattering intensity (Figure 8). Taking into account counterion condensation, one can consider by inspection of Figure 13 that CCIK is the relevant variable. The square root variation of p with a / is ~ rather well obeyed in spite of the downturn for high values of a / ~and , one can admit that, qualitatively, the behavior of p as a function of a can be explained by the model of Panyukov and Obukhov et a1.loJ1 Also the linear variation of p with rc for large a (Figure 5 ) is in agreement with eq 26. However, the model does not account easily for the weak sensitivity of p to a at high cross-link densities (cf. Figure 5 ) . An alternate explanation for the effect of a on p can be proposed on the basis of a modification of the submicroscopic structure of the gel under a change of the ionization degree, as sketched in Figure 14. Under preparation conditions the gels are slightly charged (a = Light scattering and SANS experiments have revealed in partially charged gels the presence of static spatial concentration fluctuations.15J8Jl The range of these fluctuations was found to be much smaller than in neutral gels, and this was interpreted as due to the counterion entropy penalty that would result from the formation of large i n h ~ m o g e n e i t i e s . ~ ~ , ~ ~ Also it was shown that these inhomogeneities are quite d e n ~ e . ’ ~Therefore, ,~~ one can consider in a first approximation the gel as a two-phase medium formed of flexible chains connecting dense regions (Figure 14a). In these regions, some of the chains are trapped in between fixed cross-links, while others are held together by interactions between the hydrophobic acid backbones. Upon ionization the chains rearrange themselves as far as possible. Those of the chains in the dense regions that are free to do it move apart to give a new structure as sketched in Figure 14b. Addition of salt produces a screening of the interactions, and one recovers the structure depicted in Figure 14a. The structural changes of the gels upon ionization can be monitored by scattering experiments. A peak appears in the structure factor measured by SANS.12J3J8331Also the frozen-in component of the scattered intensity, which reflects the presence of dense regions, is strongly reduced by the presence of electrical charges (Figure 8).
f l ~ .For the data corresponding to f
Macromolecules, Vol. 28, No. 1, 1995
Elastic Properties of Polyelectrolyte Gels 205 limit of experimental error. Moreover, one cannot exclude a slight change in gel architecture due to differences in chemical reactivity for charged and uncharged monomers. At swelling equilibrium, however, differences seem more visible since the gel prepared from ionized monomers has a slightly smaller shear modulus (Q, = 224 f 20, p, = 1800 & 200 Pa) than the gel neutralized after preparation (&e = 240 f 25, p, = 2300 f 200 Pa). B. Gels at the Swelling Equilibrium. To interpret the results, it is important to have some information on the screening range in the investigated gels. It seems that most of our results have been obtained in a strong screening regime as inferred from the following observations. Let us consider for instance the gels with rc = 0.02. If the cross-links were randomly distributed and if the gel was free of defects, the average number of monomeric units N in between two consecutive crosslinks would be about 25. For a gel with an ionization degree a = 0.35 swollen at equilibrium in pure water, the chains can be considered to have a stretched conformation. Taking a = 9 A for the statistical unit length and b = 2.5 A as the monomer len h, we would get an end-to-end distance of about 60 This is the same order of magnitude as K - ~in these conditions (Table 2). Thus, in pure water, we have KR, = 1for the gels at swelling equilibrium. In fact, two partly compensating errors are not taken into account in this rough estimation: chains are likely not fully stretched but, on the other hand, the effective N value is probably larger than 25 due to defects in the network (see below). Now, in the low-swelling range in the presence of a large excess of salt, the end-to-end distance of the same chain in its Gaussian conformation is reduced to about 23 A. In that case, K - . ~values are much smaller (Table 2) and the KR, value can increase up to about 10. Thus it seems that, in the whole swelling range, the inequality K R L~ 1 is always fulfilled. The same conclusions can be reached for the smaller cross-linking degrees since the increase of K - ~at swelling equilibrium in pure water (Tables 2 and 4) is compensated by an increase in N . As a matter of fact we have seen in the theoretical section that the expression (18)obtained for the strong screening regime still applies in the weak screening regime and crosses over smoothly to the case where no added salt is present (eq 10). However, one possible difficulty concerns the systems for which the number of counterions is approximately equal to that of the salt ions. In this case, the osmotic pressure resulting from the Donnan equilibrium leads to a rather complex expression for Qe, which we tentatively approximate by eq 18. In doing so we underestimate the osmotic pressure by a factor which is at the maximum 1.6 for a+ = 2&, but we cannot discard in this range of ionic strength the effect of the direct electrostatic intera c t i o n ~and ~ ~the effect of the non-Gaussian behavior of the e l a ~ t i c i t y .Another ~~ problem can arise in the other limit of very high salt content, where one expects deviations due to a nonnegligible contribution of the chains to the osmotic pressure compared to the ionic pressure. However, recent light scattering experiments performed on PAA gels show that the ionic contribution is dominant in the range of ionic strengths investigated in the present study.12J3 From the above discussion, it seems that our results have been obtained in experimental conditions where KR, > 1. Therefore the equations derived in the strong screening limit should apply with a possible restriction
R
Figure 14. Schematic picture for the evolution of gel structure in the reaction bath with the introduction of electrostatic
interactions: (a) small ionization degree; (b) large ionization degree.
Obviously, the shear modulus measured experimentally represents an averaged quantity where both individual chains and dense regions contribute. It seems quite likely that the relative weight of these contributions changes upon ionization. In fact, at high ionization degree the gel is much more homogeneous. It is significant in this respect that the shear modulus of ionized gels is a linear function of rc except at high cross-link density (Figure 5 ) . When r, is very large, the concentration fluctuations are almost totally frozen-in as shown by light scattering. This result seems to indicate that the dense regions pictured in Figure 14a have invaded the whole space so that the structure of the gel becomes insensitive to the ionization degree. This would explain why p is almost independent of a at high rc. The ionization procedure, Le., the preneutralization or postneutralization, should also affect the behavior of the shear modulus, since the extent of the frozen-in fluctuations seems to be reduced when the neutralization is performed prior to gelation, as inferred from SANS experiment^.^^ We have not made a systematic study on gels prepared according to the two ionization procedures. However, we found the following results for gels with C = 1M, r, = 0.02, and a = 0.35. For the gel prepared from ionized monomers, p = 3100 f 200 Pa, while for the gel ionized after preparation, p = 3360 f 200 Pa. Its seems that the shear modulus is slightly larger in the second case but this effect is almost at the
206 Skouri et al.
Macromolecules, Vol. 28, No. 1, 1995 a= 0.35 Cx=0.171M
a= 0 35 C5=0 171 M
:
' . 0
a 101
1
'
Figure 15. Shear modulus as a function of equilibrium swelling ratio in the representation of eq 27. Neutralization degree and salt content are kept constant while cross-linking degree is varied (same samples as in Figure 9). The straight line shows the predicted slope.
for the cases where the salt concentration is not much larger than the counterion concentration. However, since these theoretical expressions crossover smoothly to the no added salt conditions, it is tempting to still use them as approximations. We examine now successively the effect of the cross-link density and that of the electrostatic screening. Effect of the Cross-LinkDensity. If the gels were defect free and homogeneous, one would expect that rc N-l and therefore we would have a possible check by comparing the data to the predictions of eqs 18 and 22 relative to the equilibrium swelling degree and the shear modulus. In fact, it appears generally preferable to test the theory by examining the correlation between two macroscopic parameters, since this allows one to eliminate N , which is known to vary in a nonsimple way with rc because of trapped entanglements and dangling chains. By combining eqs 18 and 22, one obtains
-
ry
~
L
,
J
~
1000 Qe
Qe
Pe
_
100
100
k ~ T 6 ~ ( 4 7 d ~ ) ( a / K ) ~ & ~ '(27)
Figure 15 shows the variation of the product p e ( a / ~ ) - 2 with Qe in log-log coordinates for gels with varying cross-linking degree (Table 1). We have also drawn through the data a straight line with a slope -2. The agreement is not very good but again, the frozen-in fluctuations might affect the behavior of pe. In this respect it must be noted that these static fluctuations, even though they are reduced upon swelling, still remain strongly dependent on the cross-link density.15 Similarly, one can combine eqs 18 and 23 to eliminate the dependence on N and obtain (28) Figure 16 shows the variation of D,(a/~)-l as a function of Qe. The predicted decrease of this parameter with Qe is observed but the variation is larger than the theoretical expectation. A fit through the data would yield a much larger slope of about -1.4. From the swelling kinetics experiments one can also measure the parameter pJM,, (see Experimental Section). Since, a t swelling equilibrium, Ke = pe, one expects pJM,, = 317. This is a value slightly lower than the results reported in Table 1. Also the ratio pJM,, seems t o increase with rc. This could be explained again with the effect of the submicroscopic structure of the gels. The compressional modulus depends essentially on the entropy of counterions whereas the shear modulus
Figure 16. Cooperative diffusion coefficient as a function of equilibrium swellingratio in the representation of eq 28 (same samples as in Figure 9). The straight line corresponds to the expected slope.
might be affected by the presence of inhomogeneities. However, pJMo, values derived from swelling kinetics experiments might also be erroneous when De values are about cm2 s-l, because then collective motion of the chains proceeds at about the same rate as the restoration of the Donnan equilibrium. The same remark applies to the determination of De. This point is discussed later on. Effect of the Electrostatic Screening. Equations 18 and 22 predict that, for a given cross-link density, the equilibrium swelling degree and the shear modulus at the equilibrium follow scaling laws of the parameter C ~ K .In Figure 17 are reported sets of data relative to the swelling degree Qe for different series of gels (Tables 2-4). For a / values ~ larger than 1,the data are fitted satisfactorily in the log-log representation by straight lines with slopes close to the theoretical prediction 1.2 (eq 18). This agreement is found as well for the gels neutralized prior to synthesis as for gels neutralized afterward. Deviations from the power law behavior can be observed for a / values ~ smaller than 1and correspond in fact to deswollen gels with concentrations larger than in the reaction bath. One notices that the data relative to the smallest ionization degree (f = 0.1) are shifted upward in this representation. This might be due to small changes in the prefactor of the osmotic pressure (eq 61, which depends on the ionization degree. It can be noted that data from Figure 17a-c do collapse on a single curve when plotted on the same graph. However, this feature is only fortuitous and results from a compensationbetween varying cross-linking degree and concentration in the reaction bath to obtain about the same value for the polymerization degree N of the effective elastic strands. That parameter enters into eq 18 and plays a role as demonstrated by the fact that data from Figure 17d do not superimpose on those from Figure 17a-c. The variation of the equilibrium shear modulus pe with the parameter d~is depicted in Figure 18. Two ~ i.e. for small regimes are observed: at low a / values, ~ a minimum swelling ratios, pe decreases with a / until and then, for the larger swelling ratios, pe increases. In the whole range, the variations relative to the different ionization degrees cannot be superimposed satisfactorily . the prefactor in the osmotic as a function of a / ~ Again pressure might be responsible for that effect. At low a / ~the , shear modulus decrease can be approximated by power laws with exponents ranging between -112 and -1. The parameter d~can be eliminated between eqs 18 and 22, which amounts to eliminating the excluded volume parameter in these expressions. The resulting
Macromolecules, Vol.28, No.1, 1995
Elastic Properties of Polyelectrolyte Gels 207
1
i
615
100
I 1 ' ' '
101
~
100
d K
(A)
10'
102
0. I
0
L-
(d)
100
10-1
A
102
10'
100
-
100 101
,
100
i
i
102
a/K (A) C%/K (A) lo' Figure 17. Variation of the swelling ratio Qe as a function of a / for ~ gels with different conditions of preparation: (a) Cprep= 1.44 M,re = 0.5%,ionized after the gelation; (b) Cprep as indicated, rc = 0.47%,ionized before the gelation (data from ref 30); (c) Cprep = 1.11M, rc = 1%,ionized after the gelation; (d) Cprep = 1.11M,re = 2%,ionized after the gelation. The straight lines show the predicted 6/5 slope (eq 18). equilibrium, the excluded volume parameter being varied by addition of salt andor change of ionization degree. For a neutral gel, the C*theorem, based on the m packing c ~ n d i t i o npredicts ,~ that, under a variation of 4 the excluded volume parameter resulting from an I_" interchange of solvent, the shear modulus Pe ~ B T I R ~ ~ is inversely proportional to &e. In the case of ionized gels, even the smallest swelling ratios are already of 0 0.1 the same order of magnitude as the equilibrium swelling 0.2 ratios in neutral gels. Therefore it can be expected that 3 035 higher swelling ratios at smaller salt concentrations can be obtained only by stretching the polymer chains. At this point it is interesting to refer back to the Panyukov and Obukhov et al. approachlOJ1 and t o examine the experimental data in the light of their model. It is still assumed that swelling causes junctions to move affinely; that is, they move apart by the same linear expansion A as the macroscopic network, A = (Q$ &,#I3, with QObeing the swelling ratio in the preparation state. The end-to-end distance Re becomes ARprep under swelling, Rprepbeing the same quantity in the expresses simply the affine character of the deformation, preparation state, but the end-to-end distance in the which was assumed in the model. The experimental reference state is now taken to be that of a free chain variation of Pe as a function of &e is given in Figure 19. in a solution at concentration @e. The scaling result is Within experimental accuracy, the data are consistent used for the osmotic pressure, i.e., n,,$kBT 4'-3, where with the affine deformation assumption in the range 6 is the correlation length, and no packing assumption where p e decreases with increasing swelling ratio Qe. is used. In the case of gels prepared in the absence of Exponents of the power law Pe(Q) between 113 and 0.6 charges (0solvent) and then ionized and swollen at have been reported for a number of osmotic deswelling experiments in good solvent ~ o n d i t i o n s . ~ Values ~ * ~ ~ - ~ ~ equilibrium in the presence of a salt excess, Rprep corresponds to Gaussian strands and the end-to-end of the exponent larger than 113 were attributed to a distance in the reference state is given by eq 24. It consequence of a deinterpenetration process.23 Also, in follows that the same conditions, the model proposed by PanyukovlO and Obukhov et a1.l1 predicts an exponent of 7/12 for Qe N3/5a-9/5 ~ ~ ~ ~ ~ ) 8 3 / ~ gels in good solvents. It must, however, be remarked that these results refer to deswelling experiments in ~e kBTN24/17a-12/17Q0-9/10Ue-3/5 (30) which the excluded volume parameter remains constant, k Ta- 3 N 3/4Q0-i/2Qe-i the gels being immersed in concentrated solutions of Pe long chains. Our study concerned gels a t the swelling where the last equation is obtained by eliminating ue 1o4
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Macromolecules, Vol. 28, No. 1, 1995
208 Skouri et al. 1o4 10
t
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lod
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02
Figure 19. Shear modulus as a function of swelling equilibrium ratio (same data as in Figure 18). Continuous lines correspond to predictions from the model based on deviations from Gaussian elasticity at high Qe values8 (numbers of statistical units as indicated). An arbitrary normalization was applied to match the calculated curves to the experimental data.
Figure 20. Variation of the cooperative diffusion coefficient with equilibrium swelling ratio (data from Table 2). The straight line has the predicted slope in eq 31.
Gaussian elasticity in the high-swelling range leads us to question the validity of the derivation of the equilibrium swelling degree. In fact, the upturn of p, in the high-swelling range should lead to a smaller variation of Qe with a / ~in the same range. This is not seen in between the two first equations. Taking into account Figure 17. However, in the representation of Figure 16, eq 16, the Qe dependence on a / ~is the same as in (18) there is a partial compensation, as a downwards shift and in agreement with the data. On the other hand, of Qe is accompanied by a shift to the left of the the Qe-l dependence for pe does not agree with the parameter a / ~ . experimental behavior in Figure 19. It can also be noted Turning now to the dynamical properties, one can that the use of the scaling result for the osmotic express the equilibrium diffusion coefficient as a funcpressure is fully inconsistent with eq 21, which is supported by most experimental r e s ~ l t s . ~ How~ - ~ ~ , ~tion ~ of Qe for gels with constant N by combining eqs 18 ever, the use of eq 21 in the model does not provide any and 23. This leads to better agreement with the experiments since then even ~ for Qe is incorrect, with an exponent the a / dependence De B T ~ 2 / 3 a-1 ~ ~ 1 1 3 (31) 30117, much larger than 615. TO A related remark is that both the elastic term (eq 5) with the affine deformation hypothesis and the osmotic The variation of De with Qe is plotted in Figure 20. term (eqs 16 and 21) seem to be correctly estimated in The diffusion coefficient is found to increase with Qe as the model by Barrat et al. Thus the right prediction predicted by the theory. It must be noted that the for the a / dependence ~ of Qe would not result from the opposite behavior is found for neutral gels.27 However, cancellation of two errors as is often the case in Flory one observes a much larger variation of De than type theories.26 In this context, the shear modulus expected, with an exponent close to 1. Furthermore, the results in Figure 18 appear even more surprising. data relative to the lowest ionization degree ( a = 0.1) are shifted to lower values. In the high-Qe range the upturn in pe has already One can note that eq 31 is based on eq 23, which has been reported for gels of acrylamide-sodium acrylate been derived from eq 14, assuming a monotonous copolymers30 and hydrolyzed polyacrylamide.8 The variation for pe as a function of a / ~(eq 22). The latter results were interpreted as the onset of non-Gaussian is not obeyed for large Qe values. Therefore it is elasticity of the chains due to their highly stretched interesting to combine directly the experimental results c o n f ~ r m a t i o n . The ~ ~ strain-stress relation becomes according to eq 14, which is more general than eq 31 then nonlinear but the shear modulus can still be and does not rely on the a / dependence ~ of the experiobtained from the initial slope of the true stress variamental quantities. Combining (9) and (141, one gets tion versus the deformation function (A2 - A-1). In Figure 19 are reported the curves pe(Qe) calculated by means of the approach of ref 8 for different values of (32) the number of statistical units between two consecutive cross-links. The agreement is poor but the general behavior of the calculated curves would favor a number Figure 21 shows the log-log variation of the ratio of statistical units N about 15-20, which corresponds D$pe as a function of Qe for the series of gels whose to a number of monomers vC about 60, significantly characteristics are given in Table 2. Again one observes larger than estimated in the approximation of an ideal the small shift between the samples with a = 0.1 and gel. This means that there are many cross-links in those with higher ionization degree. Even if the data efficient in the gel either because of the presence of analysis is restricted to the lowest Qe range, a search dangling chains or because they are grouped in small for a power law behavior would yield exponents about aggregates. 1, larger than the predicted 213 value. Thus, even the validity of eq 14 seems to be questioned. It is also interesting to remark that the upturn of the A possible explanation lies in the complexity of the equilibrium shear modulus occurs for values of Qe medium that contains three Merent components: monocorresponding to a concentration of counterions apmers, counterions (Na+),and salt ions (Na+ and Cl-), proximately equal to the salt concentration. At this the small amount of H+ ions resulting from the sponstage, the chains start to be stretched. In that range, taneous dissociation of the monomers being neglected. one observes experimentally a nonlinear behavior of the This implies three eigenvalues for the matrix of the stress versus (az - A-1). The observation of a non-
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Macromolecules, Vol. 28, No. 1, 1995 loa/
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Elastic Properties of Polyelectrolyte Gels 209
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100 Qe
Figure 21. Plot of the ratio qrD$pe as a function of Qe (eq 321, qr being the ratio of solvent viscosity to pure water viscosity. The straight lines shows the prediction fiom eq 32.
relaxation rates of concentration fluctuations and thus three characteristic decay rates, which have been discussed in detail by Ajdari et a1.13,43,44 The fastest mode is the plasmon mode, which corresponds to the restoration of electroneutrality through fast nondiffusive motion of the small ions. The two remaining modes are diffusive and must participate in the kinetics of swelling. The fastest mode corresponds t o the time scales where the Donnan equilibrium is restored and is characterized by a diffusion coefficient typically loW5cm2 s-l. The slowest relaxation mode is the one we have considered up to now and is related t o the collective motion of polymer chains with electroneutrality and Donnan equilibrium being preserved.43 In fact, it can be seen in Figure 16 that the experimental values of De in the high-swelling limit are of the same order of magnitude as the diffusion coefficients of ions. Both swelling and Donnan equilibration proceed with about the same rate. Therefore the assumption that the Donnan and collective motion modes are uncoupled is likely to be nonvalid. One must also note that this coupling effect between the two modes must affect the determination of pJMos from the kinetics of swelling experiments.
Conclusion The results reported in this paper illustrate the effect of electric charges on the thermodynamic properties of gels. Moderately cross-linked gels of PAA prepared at low pH and subsequently ionized have been studied by means of different techniques: static and dynamic light scattering, kinetics of swelling, and uniaxial compression experiments. At a fixed polymer concentration, close to that of preparation, the shear modulus is found to decrease upon increasing the ionization degree. This result can be explained by assuming that the reference state used t o compute the elastic part of the free energy is changed upon ionization of the chains.lOJi However, that assumption fails to explain swelling equilibrium results. An alternate explanation takes into account the tendency of these systems to undergo microscopic phase segregation. That behavior is observed as well for PAA solutions but is coupled in the case of gels to the frozen inhomogeneities associated with the formation of the network, i.e., the radical copolymerization of cross-links and monomers with different reactivities. We believe that the elastic properties of the gels reflect complicated averages between the elastic contributions from dense and more dilute regions and that the respective weights of these contributions can be modified upon ionization and partial rearrangement of the chains, as long as the cross-linking degree is not too high. While this picture
is only conjectural and qualitative, it is supported by light scattering results that show a decrease of the frozen-in component upon ionization. Would this conjecture be correct, then this would be the first experimental evidence of a correlation between the macroscopic properties and the submicroscopic structure. The properties of gels swollen a t equilibrium in solvents with different salt contents can be described by a simple picture of gels swollen by an electrostatic excluded volume proportional to ( a / ~ ) Surprisingly, ~. the excluded volume approach in the strong screening regime and the Donnan equilibrium model in the weak screening regime lead t o the same prediction for the equilibrium swelling degree. This is probably not merely coincidental and might have some physical meaning. In fact, most of the results have been obtained in the strong screening regime. It appears that for gels formed from flexible chains the weak screening regime seems to be unattainable at any cross-link density as shown by the following observation: according to eqs 8 and 9 relative to salt-free gels Re u-1/2,$e-112a-1/4 whereas K ( 4 ~ c 1 ~ ) ~ ~so~ that , $ ~KR ~e~ ~ all4 a ~ ~is ~ , independent of N and ,$e and can be decreased only by a reduction of a,in which case other effects associated with backbone interactions come into play. The swelling equilibrium degree was found to scale like ( a / ~ whatever )~/~ the conditions of synthesis for the gels. For given cross-linking degree and concentration of preparation, the plots of &e vs a / ~lead to a good superposition of the data except for the smallest ionization degree cf= O.l), maybe due to a prefactor effect in the expression for the osmotic pressure. The 615 slope is always obeyed, in agreement with the model.g On the other hand, the shear modulus results are not well described by the model and the ( a / ~ )dependence -~~ is not clearly observed. However, as long as the equilibrium swelling degree is not too large, they are consistent with the postulated affine deformation process, which might be favored by a better homogeneity of the gels, due to the presence of electrical charges, and by the large swelling degrees that facilitate the deinterpenetration of the chains. In the limit of very high swelling degree, where the chains become strongly stretched, the shear modulus is no longer given by Gaussian elasticity and increases with the swelling ratio. Finally, measurements of the kinetics of swelling allow one to determine an effective diffusion coefficient of the gels a t the swelling equilibrium. This coefficient is found to increase with the swelling degree, contrary to what is observed for neutral gels and in qualitative agreement with the predictions of an approach based on an electrostatic excluded volume. However, the variation of the diffusion coefficient with a / or ~ with the swelling degree is much larger than predicted. This could be partly due to the coupling between the establishment of a Donnan equilibrium and the collective diffusion of the network strands, coupling that is not taken into account by the model. Several concluding remarks can be made. In neutral gels it was found that topological defects, such as dangling ends in the network, affect the shear modulus sooner than the equilibrium swelling degree.45 Thus it would be tempting t o consider that the data reported here support the picture by Barrat et al., a t least as far as only static equilibrium properties are concerned, the departure of shear modulus results from the prediction being attributed to the complex structure of these gels.
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210 Skouri et al.
However, it must recognized that such a simple picture makes no room for the variation of persistence length with neutralization and salt content, a problem still under debate t ~ d a y . In ~ ~this , ~context, ~ the success of eq 18 in describing the equilibrium ratio of our gels might be only fortuitous. Anyway this equation provided a very useful hint for a new powerful representation of these data. A theoretical justification for this phenomenological equation may still need to be found. Acknowledgment. We are grateful to J. F. Joanny and P. Pincus for continuous interactions during this work. Helpful discussions with M. Rubinstein are also gratefully acknowledged. References and Notes (1) Katchalsky, A.; Lifson, S.;Heisenberg, H. J . Polym. Sci. 1951, 7, 571. (2) Katchalsky, A.; Michaeli, J. J. Polym. Sci. 1955, 9, 69. (3) Flory, P. J . The Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (4) Hasa, J.;Ilavsky, M.; Dusek, K. J . Polym. Sci. 1975,13,253. (5) Ricka, J.; Tanaka, T. Macromolecules 1984, 17, 2916. (6) Brannon-Peppas, L.;Peppas, N. A. Polym. Bull. 1988,20,285. (7) Konak, C.; Bansil, R. Polymer 1989,30, 677. (8) Schroder, U. P.; Oppermann, W. Makromol. Chem., Macromol. Symp. 1993, 73, 63. (9) Barrat, J. L.; Joanny, J. F.; Pincus, P. J . Phys. 11Fr. 1992, 2, 1531. (10) Panyukov, S. V. Sou. Phys.-JETP (Engl.Transl.) 1990, 71, 372. (11) Obukhov, S. P.; Rubinstein, M.; Colby, R. H., preprint. (12) Schosseler, F.; Ilmain, F.; Candau, S. J. Macromolecules 1991, 24, 225. (13) Schosseler, F.; Moussaid, A.; Munch, J. P.; Candau, S. J. J . Phys. 11 Fr. 1991, I, 1197. Moussaid, A.; Munch, J. P.; Schosseler, F.; Candau, S. J. J . Phys. 11Fr. 1991, 1, 637. Skouri, R.; Munch, J. P.; Schosseler, F.; Candau, S. J. Europhys. Lett. 1993,23, 635. Joosten, J. G. H.; McCarthy, J. L.; Pusey, P. N. Macromolecules 1991,24, 6690. Pusev. P. N.: Van Meeen. W. Phvsica A 1989.157. 705. (18) Mou&id, A.; Candau,-S. J.;Joosten, J. G. H. Macromolecules 1994,27, 2102. (19) Li, Y.; Tanaka, T. J . Chem. Phys. 1990, 92, 1365.
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